Linear Methods in the Proof of Jackson Type Inequalities and Applications to Estimates of Functionals with Two Known Moments
- 35 Downloads
We propose a new method for estimating functionals in terms of higher order moduli of continuity with explicit constants. Using this method, we estimate deviations of linear methods of approximation by entire functions of finite degree and, in particular, by trigonometric polynomials. For illustration of the results, we derive estimates for the Riesz and Akhiezer–Krein–Favard averages. Bibliography: 14 titles.
KeywordsEntire Function Trigonometric Polynomial Conjugate Function Primitive Function Riesz Operator
Unable to display preview. Download preview PDF.
- 1.O. L. Vinogradov and V. V. Zhuk, “Estimates for functionals with a known finite set of moments in terms of moduli of continuity and behavior of constants in the Jackson-type inequalities” Algebra Anal. 24, No. 5, 1–43 (2012); English transl.: St. Petersbg. Math. J. 24, No. 5, 691–721 (2013).CrossRefzbMATHMathSciNetGoogle Scholar
- 2.A. F. Timan, Theory of Approximations of Functions of Real Variable [in Russian], Fizmatgiz, Moscow (1960).Google Scholar
- 3.A. I. Stepanets, Methods of Approximation Theory, Leiden (2005).Google Scholar
- 4.V. V. Zhuk, Approximation of Periodic Functions [in Russian], Leningr. Univ. Press, Leningr. (1982).Google Scholar
- 5.V. V. Zhuk, STructural Properties of Functions and Accuracy of Approximation [in Russian], Leningr. Univ. Press, Leningr. (1984).Google Scholar
- 6.N. I. Akhiezer, Lectures on Approximation Theory [in Russian], Nauka, Moscow (1965).Google Scholar
- 8.B. M. Levitan, Almost-Periodic Functions [in Russian], GITL, Moscow (1953).Google Scholar
- 12.V. V. Zhuk and G. I. Natanson, Trigonometric Fourier Series and Elements of Approximation Theory [in Russian], Leningr. Univ. Press, Leningr. (1983).Google Scholar
- 13.V. V. Zhuk and S. Yu. Pimenov, “On norms of the Akhiezer–Krein–Favard sums” [in Russian], Vestn. St. Peterb. Gos. Univ., Ser. 10, No. 4, 37–47 (2006).Google Scholar